ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Applications of Derivatives (Set 5): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review.…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. The tangents drawn from the origin to the parabola y = x² + 1 have slopes:
Answer: C. ±2
Explanation: Tangent at (c, c² + 1) has slope 2c; passing through the origin needs c² + 1 = 2c², so c = ±1 and the slopes are ±2.
Q2. A right circular cylinder is inscribed in a sphere of radius √3. Its greatest possible volume is:
Answer: C. 4π
Explanation: With height h, r² = 3 − h²/4 and V = π(3h − h³/4). V′ = 0 gives h = 2, so V = π(6 − 2) = 4π.
Q3. The largest rectangle that can be inscribed in a semicircle of radius 5 (one side on the diameter) has area:
Answer: A. 25
Explanation: With half-base x and height y, x² + y² = 25 and area 2xy ≤ x² + y² = 25, with equality at x = y.
Q4. The least distance from the point (0, 3) to the parabola y = x² is:
Answer: C. √11/2
Explanation: With w = x² ≥ 0, d² = w + (w − 3)² = w² − 5w + 9, minimised at w = 5/2 giving d² = 11/4.
Q5. A wire of length 12 cm is cut into two pieces and each piece is bent into a square. The least possible total area of the two squares is:
Answer: A. 9/2 cm²
Explanation: With pieces p and 12 − p, area = (p² + (12 − p)²)/16, minimum at p = 6: 72/16 = 9/2.
Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.
Q6. A lamp is 6 m above the ground. A 1.5 m tall person walks away from the point below the lamp at 1.2 m/s. The tip of the shadow moves at:
Q7. A balloon rises vertically at 3 m/s from a point 40 m from an observer. When it is 30 m high, the angle of elevation of the balloon is increasing at:
Q8. The surface area of a sphere grows at 8π cm²/s. When the radius is 2 cm, the volume is increasing at:
Q9. For x > 0, f(x) = x ln x is decreasing on:
Q10. The graph of f(x) = ln(1 + x²) is concave downward for:
Q11. The maximum value of 2 sin x + sin 2x on [0, π/2] is:
Q12. For f(x) = √x on [1, 9], the value of c in (1, 9) given by the mean value theorem is:
Q13. In the Maclaurin series of tan⁻¹x, the coefficient of x³ is:
Q14. In the Maclaurin series of e^(−x²), the coefficient of x⁴ is:
Q15. In the Maclaurin series of eˣ cos x, the coefficient of x³ is:
Q16. In the Taylor series of √x about x = 4, the coefficient of (x − 4)² is:
Q17. The radius of a sphere is measured as 5 cm with a possible error of 0.02 cm. The approximate error in its volume is:
Q18. The edge of a cube is measured with a 1.5% error. The approximate percentage error in its volume is:
Q19. A ball is thrown upward so that its height is h = 20t − 5t² metres. Its speed when it passes the height 15 m on the way down is:
Q20. A closed box with a square base must have volume 64 cm³. Its least possible surface area is:
Q21. A cylinder has radius r and height h. When r = 4 cm and h = 5 cm, r is increasing at 2 cm/s and h is decreasing at 3 cm/s. The volume is changing at:
Q22. The number of critical points of f(x) = x + 2 sin x in (0, 2π) is:
Q23. The cubic x³ − 6x² + 9x + k has three distinct real roots exactly when:
Q24. The curve y = x eˣ has a point of inflection at x equal to:
Q25. If x, y > 0 and x + y = 9, the maximum value of x²y is:
Q26. lim (x→0) (eˣ − 1 − x)/(1 − cos x) equals:
Q27. The function f(x) = x³ + kx² + 3x is increasing for all real x when:
Q28. The normal to y = x² at (1, 1) meets the parabola again at the point whose x-coordinate is:
Q29. The tangent to y = 1/x at the point (2, 1/2) and the coordinate axes form a triangle of area:
Q30. The point on the line y = 2x + 3 nearest to the origin is: